On Triple Ratios: 6 of 6

Complex Multiplications

Given an arithmetic of dual numbers x+ey, and a conjugation # that turns e into –e, then consider this operation:

x*y   =  ( x*y + (#x)*y + x*(#y) – (#x)*(#y) ) / 2

-         where * is dual-arithmetic multiplication: e*e = 1.

Then these laws follow:

* is commutative

* distributes over addition

If r is real, then x*(r*y)  = r *(x*y)

1*1 = 1

e*e = -1

From which follows:

* is isomorphic to complex multiplication.

Here are three complex multiplications of triple ratios:

x*₁y   =  ( x*y +₁ (1#x)*y +₁ x*(1#y) –₁ (1#x)*(1#y) ) / 2₁

x*₂y   =  ( x*y +₂ (2#x)*y +₂ x*(2#y) –₂ (2#x)*(2#y) ) / 2₂

x*₃y   =  ( x*y +₃ (3#x)*y +₃ x*(3#y) –₃ (3#x)*(3#y) ) / 2₃

-         where 2₁  =  (1;2;2)    ;   2₂  =  (2;1;2)   ;   2₃  =  (2;2;1)

Exercise for the ambitious student:

Find formulas for x*₁y in terms of x₁, x₂, x₃, y₁, y₂, and y₃; and likewise for x*₂y  and x*₃y.

          Open Questions

          How do the three arithmetics relate, other than by S₃ conjugation? Are there identities involving products of additions, reductions, or complex multiplications?

          Define, for any triple ratios A = (a₁, a₂, a₃)  and B = (b₁, b₂, b₃) , the two functions M and m:

          M(A,B)      =       (A+₁B)*(A+₂B)*(A+₃B) / (A*B)

                             =   1  /  ( a₂b₃ + a₃b₂  ;  a₁b₃ + a₃b₁ ;  a₁b₂ + a₂b₁ )

          m(A,B)       =       (A[+]₁B)*(A[+]₂B)*(A[+]₃B) / (A*B)

                             =   (a₁b₁(a₂b₃+a₃b₂) ; a₂b₂(a₁b₃+a₃b₁) ; a₃b₃(a₁b₂+a₂b₁))

          Since (x[+]_ay)*(x+_ay)   = x*y, it follows that:

                   M(A,B)*m(a,b)             =       A*B

                   m(a,b)                  =       A*B / M(A,B)    

1/m(A,B)             =       M(1/A, 1/B)

1/M(A,B)            =       m(1/A, 1/B)

For any triple ratios A, B and C:

M(A,A)                =       A

m(A,A)                =       A

A * M(B,C)         =       M(A*B, A*C)

A * m(B,C)         =       m(A*B, A*C)

A / M(B,C) =       m(A/B, A/C)

A / m(B,C) =       M(A/B, A/C)

          And for any index a:

                   a# M(A,B)           =       M(a#A, a#B)

                   a# m(A,B)           =       m(a#A, a#B)

M and m are symmetric under S₃, like times and divide.

How do M and m relate to the three additions?